The Monty Hall Problem: stick or switch?

The Monty Hall Problem is nice brain teaser, based on the American television game show Let's Make a Deal and named after its original host, Monty Hall.

It became famous as a question from a reader's letter quoted in Marilyn vos Savant's "Ask Marilyn" column in Parade magazine in 1990 (Internet Archive [1])

Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, 
which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? 

The answer is YOU SHOULD ALWAYS SWITCH!

Crazy eh? Not really if you think about it in basic simple probabilistic terms. 
Let's try.

At the beginning of the Game, each door is equally probable to hide the car, so 1/3. 
We really cannot do much about it, we don't know neither can't do any better.

The player picks a door (let's say door 1) and has 1/3 probability to win and 2/3 to lose (there are still two doors, each one with 1/3 probability to hide the car).
In other terms, the player is likely to loose (1/3 Vs. 2/3).

(Source: Wikipedia)

Now, when the host opens one (door 3 in the picture) of the two remaining doors (which combined probability to hide the prize was and is still 2/3!!!!), we have more information!!!

Of the original 2/3 combined probability of door 2 and door 3 we now know that actually door 3 hid a goat, this means that its probability to hide the car is well.... 0, just zero! 
As a consequence, door 2 gets the FULL 2/3 probability that we originally assigned to door 2 and door 3....  

(source: Wikipedia)

Now, if you decide to switch and go for door 2, you have 2/3 probability to win!!!!

Still unconvinced?
Well, just scale the same problem/game up to one having 100 doors.

At the beginning you choose a door out of 100, so you have 1/100 probability to win or 99/100 to loose. 
When the host asks you, will you switch? Probably yes! 

After the host opens 98 doors, your original choice has still 1/100 chance to win, but the one door that the host did not open has now 99/100 chance to contain the prize!

If you switch you are very likely gonna win!!!! After all 99/100 is much bigger than 1/100, 99-bigger ;)

The problem is quite famous because the original answer by Marilyn vos Savant caused a lot of debate, the Vox video [3] makes a nice job in summarizing the controversies.  
What really stuck me, is the fact that Marilyn vos Savant received many many angry letters (thousands of them!!!) by academics who refused to accept her solution (see [1][2][3])!!! 
Can you believe this????

I highly recommend you to read the Wikipedia Page about the Monty Hall Problem [2] :)

A nice summary




Another nice summary

With Conditional Probability





Bonus
References to the Monty Hall Problem can be found in:


Sources



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